Electronic Communications in Probability

Uniform bounds for exponential moment of maximum of a Dyck path

Oleksiy Khorunzhiy and Jean-François Marckert

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Let us consider the maximum $M(D)$ of a Dyck path $D$ chosen uniformly in the set of Dyck paths with $2n$ steps. We prove that the exponential moment of $M(D)$ normalized by the square root of $n$ is bounded in the limit of infinite $n$. This uniform bound justifies an assumption used in literature to prove certain estimates of high moments of large random matrices.

Article information

Electron. Commun. Probab., Volume 14 (2009), paper no. 32, 327-333.

Accepted: 12 August 2009
First available in Project Euclid: 6 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability
Secondary: 60G70: Extreme value theory; extremal processes 60F99: None of the above, but in this section

Dyck paths Bernoulli bridge random matrices

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Khorunzhiy, Oleksiy; Marckert, Jean-François. Uniform bounds for exponential moment of maximum of a Dyck path. Electron. Commun. Probab. 14 (2009), paper no. 32, 327--333. doi:10.1214/ECP.v14-1486. https://projecteuclid.org/euclid.ecp/1465234741

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  • Billingsley, Patrick. Probability and measure.Second edition.Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons, Inc., New York, 1986. xiv+622 pp. ISBN: 0-471-80478-9
  • Chung, Kai Lai. Maxima in Brownian excursions. Bull. Amer. Math. Soc. 81 (1975), 742–745.
  • Dvoretzky, A.; Motzkin, Th. A problem of arrangements. Duke Math. J. 14, (1947). 305–313.
  • Feller, William. An introduction to probability theory and its applications. Vol. I.Third edition John Wiley & Sons, Inc., New York-London-Sydney 1968 xviii+509 pp.
  • Flajolet, Philippe; Odlyzko, Andrew. The average height of binary trees and other simple trees. J. Comput. System Sci. 25 (1982), no. 2, 171–213.
  • Janson, Svante; Marckert, Jean-François. Convergence of discrete snakes. J. Theoret. Probab. 18 (2005), no. 3, 615–647.
  • Kaigh, W. D. An invariance principle for random walk conditioned by a late return to zero. Ann. Probability 4 (1976), no. 1, 115–121.
  • bibitem O. Khorunzhiy, V. Vengerovsky. Ewen walks and estimates of high moments of large Wigner random matrices. Preprint arXiv:0806.0157.
  • Raney, George N. Functional composition patterns and power series reversion. Trans. Amer. Math. Soc. 94 1960 441–451.
  • Stanley, Richard P. Enumerative combinatorics. Vol. 2.With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin.Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999. xii+581 pp. ISBN: 0-521-56069-1; 0-521-78987-7
  • Pitman, J. Combinatorial stochastic processes.Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002.With a foreword by Jean Picard.Lecture Notes in Mathematics, 1875. Springer-Verlag, Berlin, 2006. x+256 pp. ISBN: 978-3-540-30990-1; 3-540-30990-X
  • SinaÄ­, Ya. G.; Soshnikov, A. B. A refinement of Wigner's semicircle law in a neighborhood of the spectrum edge for random symmetric matrices.(Russian) Funktsional. Anal. i Prilozhen. 32 (1998), no. 2, 56–79, 96; translation in Funct. Anal. Appl. 32 (1998), no. 2, 114–131
  • Smith, Laurel; Diaconis, Persi. Honest Bernoulli excursions. J. Appl. Probab. 25 (1988), no. 3, 464–477.
  • Soshnikov, Alexander. Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207 (1999), no. 3, 697–733.
  • Wigner, Eugene P. Characteristic vectors of bordered matrices with infinite dimensions. Ann. of Math. (2) 62 (1955), 548–564.